Factorial Design Variations

Here, we'll look at a number of different factorial designs. We'll begin with a
two-factor design where one of the factors has more than two levels. Then we'll introduce
the three-factor design. Finally, we'll present the idea of the incomplete factorial
design.

A 2x3 Example

For these examples, let's construct an example where we wish to study of the effect of
different treatment combinations for cocaine abuse. Here, the dependent measure is
severity of illness rating done by the treatment staff. The outcome ranges from 1 to 10
where higher scores indicate more severe illness: in this case, more severe cocaine
addiction. Furthermore, assume that the levels of treatment are:

Factor 1: Treatment

psychotherapy

behavior modification

Factor 2: Setting

inpatient

day treatment

outpatient

Note that the
setting factor in this example has three levels.

The first figure shows what an effect for setting outcome might look like. You have to
be very careful in interpreting these results because higher scores mean the patient is
doing worse. It's clear that inpatient treatment works best, day treatment is next
best, and outpatient treatment is worst of the three. It's also clear that there is no
difference between the two treatment levels (psychotherapy and behavior modification).
Even though both graphs in the figure depict the exact same data, I think it's easier to
see the main effect for setting in the graph on the lower left where setting is depicted
with different lines on the graph rather than at different points along the horizontal
axis.

The second figure shows a main effect for treatment with psychotherapy performing
better (remember the direction of the outcome variable) in all settings than behavior
modification. The effect is clearer in the graph on the lower right where treatment levels
are used for the lines. Note
that in both this and the previous figure the lines in all graphs are parallel indicating
that there are no interaction effects.

Now, let's look at a few of the possible interaction effects. In the first case, we
see that day treatment is never the best condition. Furthermore, we see that psychotherapy
works best with inpatient care and behavior modification works best with outpatient care.

The other interaction effect example is a bit more complicated. Although there may be
some main effects mixed in with the interaction, what's important here is that there is a
unique combination of levels of factors that stands out as superior: psychotherapy done in
the inpatient setting. Once
we identify a "best" combination like this, it is almost irrelevant what is
going on with main effects.

A Three-Factor Example

Now let's examine what a three-factor study might look like. We'll use the same factors
as above for the first two factors. But here we'll include a new factor for dosage that
has two levels. The factor structure in this 2 x 2 x 3 factorial experiment is:

Factor 1: Dosage

100 mg.

300 mg.

Factor 2: Treatment

psychotherapy

behavior modification

Factor 3: Setting

inpatient

day treatment

outpatient

Notice that in this
design we have 2x2x3=12 groups! Although it's tempting in factorial studies to add more
factors, the number of groups always increases multiplicatively (is that a real word?).
Notice also that in order to even show the tables of means we have to have to tables that
each show a two factor relationship. It's also difficult to graph the results in a study
like this because there will be a large number of different possible graphs. In the
statistical analysis you can look at the main effects for each of your three factors, can
look at the three two-way interactions (e.g., treatment vs. dosage, treatment vs. setting,
and setting vs. dosage) and you can look at the one three-way interaction. Whatever else
may be happening, it is clear that one combination of three levels works best: 300 mg. and
psychotherapy in an inpatient setting. Thus, we have a three-way interaction in this
study. If you were an administrator having to make a choice among the different treatment
combinations you would be best advised to select that one (assuming your patients and
setting are comparable to the ones in this study).

Incomplete Factorial Design

It's clear that factorial designs can become cumbersome and have too many groups even
with only a few factors. In much research, you won't be interested in a fully-crossed
factorial design like the ones we've been showing that pair every combination of
levels of factors. Some of the combinations may not make sense from a policy or
administrative perspective, or you simply may not have enough funds to implement all
combinations. In this case, you may decide to implement an incomplete factorial design. In
this variation, some of the cells are intentionally left empty -- you don't assign people
to get those combinations of factors.

One of the most common uses of incomplete factorial design is to allow for a control or
placebo group that receives no treatment. In this case, it is actually impossible to
implement a group that simultaneously has several levels of treatment factors and receives
no treatment at all. So, we consider the control group to be its own cell in an incomplete
factorial rubric (as shown in the figure). This allows us to conduct both relative and
absolute treatment comparisons within a single study and to get a fairly precise look at
different treatment combinations.